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<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"><meta http-equiv="X-UA-Compatible" content="IE=edge,IE=9,chrome=1"><meta name="generator" content="MATLAB 2021b"><title>WBR腿部五连杆机构分析</title><style type="text/css">.rtcContent { padding: 30px; } .S0 { margin: 3px 10px 5px 4px; padding: 0px; line-height: 28.8px; min-height: 0px; white-space: pre-wrap; color: rgb(213, 80, 0); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 24px; font-weight: 400; text-align: left;  }
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.S5 { margin: 2px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(0, 0, 0); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left;  }
.S6 { margin: 10px 10px 9px 4px; padding: 0px; line-height: 21px; min-height: 0px; white-space: pre-wrap; color: rgb(0, 0, 0); font-family: Helvetica, Arial, sans-serif; font-style: normal; font-size: 14px; font-weight: 400; text-align: left;  }</style></head><body><div class = rtcContent><h1  class = 'S0' id = 'T_436425E5' ><span>WBR腿部五连杆机构分析</span></h1><div  class = 'S1'><div  class = 'S2'><span style=' font-weight: bold;'>目录</span></div><div  class = 'S3'><a href = "#H_85630354"><span>1 参数定义
</span></a><a href = "#H_924825AC"><span>2 逆运动学
</span></a><a href = "#H_4B517E59"><span>3 正运动学
</span></a><a href = "#H_25E42D93"><span>4 雅可比矩阵
</span></a><a href = "#H_431A4DCD"><span>5 动态静力学
</span></a><a href = "#H_8C28DA28"><span>6 动力学参数
</span></a><a href = "#H_53CCEAE5"><span>附录1 运动学验证</span></a></div></div><h2  class = 'S4' id = 'H_85630354' ><span>1 参数定义</span></h2><div  class = 'S5' id = 'H_CF1A4489' ><img class = "imageNode" src = "" width = "201" height = "160" alt = "leg_1.jpg" style = "vertical-align: baseline; width: 201px; height: 160px;"></img></div><h2  class = 'S4' id = 'H_924825AC' ><span>2 逆运动学</span></h2><div  class = 'S5' id = 'H_16416C46' ><span>逆运动学问题：</span><span texencoding="(\varphi_1,\varphi_2)=f(l,\theta)" style="vertical-align:-6px"><img src="" width="108.5" height="19.5" /></span></div><div  class = 'S5'><span texencoding="l_1=\sqrt{l_{1,a}^2+l^2+2l_{1,a}lsin\theta}" style="vertical-align:-8px"><img src="" width="159.5" height="26.5" /></span><span>    (2.1)</span></div><div  class = 'S5'><span texencoding="l_2=\sqrt{l_{2,a}^2+l^2-2l_{2,a}lsin\theta}" style="vertical-align:-8px"><img src="" width="159.5" height="26.5" /></span><span>    (2.2)</span></div><div  class = 'S5'><span texencoding="\varphi_1=acos\frac{l_1^2+l_{1,a}^2-l^2}{2l_1l_{1,a}}+acos\frac{l_1^2+l_{1,u}^2-l_{1,d}^2}{2l_1l_{1,u}}" style="vertical-align:-17px"><img src="" width="258.5" height="42" /></span><span>    (2.3)</span></div><div  class = 'S5'><span texencoding="\varphi_2=acos\frac{l_2^2+l_{2,a}^2-l^2}{2l_2l_{2,a}}+acos\frac{l_2^2+l_{2,u}^2-l_{2,d}^2}{2l_2l_{2,u}}" style="vertical-align:-17px"><img src="" width="258.5" height="42" /></span><span>    (2.4)</span></div><h2  class = 'S4' id = 'H_4B517E59' ><span>3 正运动学</span></h2><div  class = 'S5'><span>正运动学问题：</span><span texencoding="(l,\theta_l)=f^{-1}(\varphi_1,\varphi_2)" style="vertical-align:-6px"><img src="" width="122.5" height="20.5" /></span></div><div  class = 'S5'><span texencoding="(l_{1,a}-l_{1,u}cos\varphi_1+lsin\theta)^2+(l_{1,u}sin\varphi_1-lcos\theta)^2=l_{1,d}^2" style="vertical-align:-8px"><img src="" width="318" height="22.5" /></span><span>    (3.1)</span></div><div  class = 'S5'><span texencoding="(l_{2,a}-l_{2,u}cos\varphi_2-lsin\theta)^2+(l_{2,u}sin\varphi_2-lcos\theta)^2=l_{2,d}^2" style="vertical-align:-8px"><img src="" width="318" height="22.5" /></span><span>    (3.2)</span></div><div  class = 'S5'><span>令 </span><span texencoding="x_e = lsin\theta" style="vertical-align:-6px"><img src="" width="62.5" height="19.5" /></span><span> , </span><span texencoding="y_e=lcos\theta" style="vertical-align:-6px"><img src="" width="65" height="19.5" /></span><span>，记 </span><span texencoding="x_i=l_{i,a}-l_{i,u}cos\varphi_i" style="vertical-align:-6px"><img src="" width="109" height="19.5" /></span><span> ,  </span><span texencoding="y_i = l_{i,u}sin\varphi_i" style="vertical-align:-6px"><img src="" width="75" height="19.5" /></span><span> , </span><span texencoding="i=1,2" style="vertical-align:-5px"><img src="" width="47" height="17.5" /></span></div><div  class = 'S5'><span>方程组 (3.1)(3.2) 改写为</span></div><div  class = 'S5'><span texencoding="(x_e+x_1)^2+(y_e-y_1)^2=l_{1,d}^2" style="vertical-align:-8px"><img src="" width="168.5" height="22.5" /></span><span>    (3.3)</span></div><div  class = 'S5'><span texencoding="(x_e-x_2)^2+(y_e-y_2)^2=l_{2,d}^2" style="vertical-align:-8px"><img src="" width="168.5" height="22.5" /></span><span>    (3.4)</span></div><div  class = 'S6'><span>求解结果</span></div><div  class = 'S5'><span texencoding="x_e=
\frac{y_{1}\sigma_1-y_{2}\sigma_1+{l_{1,d}}^2\,x_{1}+{l_{1,d}}^2\,x_{2}-{l_{2,d}}^2\,x_{1}-{l_{2,d}}^2\,x_{2}+x_{1}\,{x_{2}}^2-{x_{1}}^2\,x_{2}-x_{1}\,{y_{1}}^2-x_{1}\,{y_{2}}^2+x_{2}\,{y_{1}}^2+x_{2}\,{y_{2}}^2-{x_{1}}^3+{x_{2}}^3+2\,x_{1}\,y_{1}\,y_{2}-2\,x_{2}\,y_{1}\,y_{2}}{2\,\left({x_{1}}^2+2\,x_{1}\,x_{2}+{x_{2}}^2+{y_{1}}^2-2\,y_{1}\,y_{2}+{y_{2}}^2\right)}" style="vertical-align:-18px"><img src="" width="790.5" height="42" /></span></div><div  class = 'S5'><span texencoding="y_e=
\frac{x_{1}\sigma_1+x_{2}\sigma_1-{l_{1,d}}^2\,y_{1}+{l_{1,d}}^2\,y_{2}+{l_{2,d}}^2\,y_{1}-{l_{2,d}}^2\,y_{2}+{x_{1}}^2\,y_{1}+{x_{1}}^2\,y_{2}+{x_{2}}^2\,y_{1}+{x_{2}}^2\,y_{2}-y_{1}\,{y_{2}}^2-{y_{1}}^2\,y_{2}+{y_{1}}^3+{y_{2}}^3+2\,x_{1}\,x_{2}\,y_{1}+2\,x_{1}\,x_{2}\,y_{2}}{2\,\left({x_{1}}^2+2\,x_{1}\,x_{2}+{x_{2}}^2+{y_{1}}^2-2\,y_{1}\,y_{2}+{y_{2}}^2\right)}" style="vertical-align:-18px"><img src="" width="790.5" height="42" /></span></div><div  class = 'S5'><span>其中</span></div><div  class = 'S5'><span texencoding="\sigma_1=
\sqrt{-{\left(-{l_{1,d} }^2 -2\,l_{1,d} \,l_{2,d} -{l_{2,d} }^2 +{x_1 }^2 +2\,x_1 \,x_2 +{x_2 }^2 +{y_1 }^2 -2\,y_1 \,y_2 +{y_2 }^2 \right)}\,{\left(-{l_{1,d} }^2 +2\,l_{1,d} \,l_{2,d} -{l_{2,d} }^2 +{x_1 }^2 +2\,x_1 \,x_2 +{x_2 }^2 +{y_1 }^2 -2\,y_1 \,y_2 +{y_2 }^2 \right)}}" style="vertical-align:-6px"><img src="" width="833" height="25" /></span></div><h2  class = 'S4' id = 'H_25E42D93' ><span>4 雅可比矩阵</span></h2><div  class = 'S5'><img class = "imageNode" src = "" width = "254" height = "160" alt = "leg_2.jpg" style = "vertical-align: baseline; width: 254px; height: 160px;"></img></div><div  class = 'S5'><span>直接对式 (2.3)(2.4) 或方程组 (3.1)(3.2) 的解求偏导的结果非常复杂，而根据速度关系可以得到表达式相对简单的雅可比矩阵，即</span></div><div  class = 'S5'><span texencoding="\dot{q}=J^{-1}\dot{p}" style="vertical-align:-5px"><img src="" width="59" height="19.5" /></span><span>    (4.1)</span></div><div  class = 'S5'><span>其中，</span><span texencoding="q=\left[\matrix{
\varphi_1 &amp; \varphi_2
}\right]^T" style="vertical-align:-7px"><img src="" width="89" height="21" /></span><span>，</span><span texencoding="p=\left[\matrix{
l &amp; \theta
}\right]^T" style="vertical-align:-5px"><img src="" width="72" height="18.5" /></span></div><div  class = 'S5'><span>令 </span><span texencoding="x_e = lsin\theta" style="vertical-align:-6px"><img src="" width="62.5" height="19.5" /></span><span> , </span><span texencoding="y_e=lcos\theta" style="vertical-align:-6px"><img src="" width="65" height="19.5" /></span><span>，记 </span><span texencoding="x_i=l_{i,a}-l_{i,u}cos\varphi_i" style="vertical-align:-6px"><img src="" width="109" height="19.5" /></span><span> ,  </span><span texencoding="y_i = l_{i,u}sin\varphi_i" style="vertical-align:-6px"><img src="" width="75" height="19.5" /></span><span> , </span><span texencoding="i=1,2" style="vertical-align:-5px"><img src="" width="47" height="17.5" /></span></div><div  class = 'S5'><span>速度约束方程</span></div><div  class = 'S5'><span texencoding="\left(\dot\varphi_1l_{1,u}\left[\matrix{-sin\varphi_1 \cr cos\varphi_1}\right]\right)^T\left[\matrix{x_e+x_1 \cr y_e-y_1}\right]
=
\left(\dot l\left[\matrix{sin\theta \cr cos\theta}\right]+\dot\theta l\left[\matrix{cos\theta \cr -sin\theta}\right]\right)^T\left[\matrix{x_e+x_1 \cr y_e-y_1}\right]" style="vertical-align:-17px"><img src="" width="408.5" height="48" /></span><span>    (4.2)</span></div><div  class = 'S5'><span texencoding="\left(\dot\varphi_2l_{2,u}\left[\matrix{sin\varphi_2 \cr cos\varphi_2}\right]\right)^T\left[\matrix{x_e-x_2 \cr y_e-y_2}\right]
=
\left(\dot l\left[\matrix{sin\theta \cr cos\theta}\right]+\dot\theta l\left[\matrix{cos\theta \cr -sin\theta}\right]\right)^T\left[\matrix{x_e-x_2 \cr y_e-y_2}\right]" style="vertical-align:-17px"><img src="" width="400.5" height="48" /></span><span>    (4.3)</span></div><div  class = 'S5'><span>由式 (4.2)(4.3) 可得</span></div><div  class = 'S5'><span texencoding="\frac{\partial\varphi_1}{\partial l}=
\frac{1}{l_{1,u}}\frac{(x_e+x_1)sin\theta+(y_e-y_1)cos\theta}{-(x_e+x_1)sin\varphi_1+(y_e-y_1)cos\varphi_1}" style="vertical-align:-17px"><img src="" width="258.5" height="40" /></span><span>    (4.4)</span></div><div  class = 'S5'><span texencoding="\frac{\partial\varphi_1}{\partial\theta}=
\frac{l}{l_{1,u}}\frac{(x_e+x_1)cos\theta-(y_e-y_1)sin\theta}{-(x_e+x_1)sin\varphi_1+(y_e-y_1)cos\varphi_1}" style="vertical-align:-17px"><img src="" width="258.5" height="40" /></span><span>    (4.5)</span></div><div  class = 'S5'><span texencoding="\frac{\partial\varphi_2}{\partial l}=
\frac{1}{l_{2,u}}\frac{(x_e-x_2)sin\theta+(y_e-y_2)cos\theta}{(x_e-x_2)sin\varphi_2+(y_e-y_2)cos\varphi_2}" style="vertical-align:-17px"><img src="" width="248" height="40" /></span><span>    (4.6)</span></div><div  class = 'S5'><span texencoding="\frac{\partial\varphi_2}{\partial\theta}=
\frac{l}{l_{2,u}}\frac{(x_e-x_2)cos\theta-(y_e-y_2)sin\theta}{(x_e-x_2)sin\varphi_2+(y_e-y_2)cos\varphi_2}" style="vertical-align:-17px"><img src="" width="248" height="40" /></span><span>    (4.7)</span></div><div  class = 'S5'><span texencoding="J=\left[\matrix{
\frac{\partial\varphi_1}{\partial l} &amp; \frac{\partial\varphi_1}{\partial\theta} \cr \frac{\partial\varphi_2}{\partial l} &amp; \frac{\partial\varphi_2}{\partial\theta}
}\right]^{-1}" style="vertical-align:-35px"><img src="" width="117" height="84" /></span><span>    (4.8)</span></div><h2  class = 'S4' id = 'H_431A4DCD' ><span>5 动态静力学</span></h2><div  class = 'S5'><img class = "imageNode" src = "" width = "250" height = "104" alt = "leg_3.jpg" style = "vertical-align: baseline; width: 250px; height: 104px;"></img></div><div  class = 'S5'><span>记 </span><span texencoding="\gamma=\left[\matrix{F_{bl} &amp; T_{bl}}\right]^T" style="vertical-align:-7px"><img src="" width="92.5" height="21" /></span><span>，</span><span texencoding="\tau=\left[\matrix{\tau_1 &amp; \tau_2}\right]^T" style="vertical-align:-7px"><img src="" width="81" height="21" /></span></div><div  class = 'S5'><span>根据虚功原理可得</span></div><div  class = 'S5'><span texencoding="\tau=J^T\gamma" style="vertical-align:-5px"><img src="" width="50" height="18.5" /></span><span>    (5.1)</span></div><h2  class = 'S4' id = 'H_8C28DA28' ><span>6 动力学参数</span></h2><div  class = 'S5'><span texencoding="\theta_l=\theta+\theta_b" style="vertical-align:-6px"><img src="" width="68.5" height="19.5" /></span><span>    (6.1)</span></div><div  class = 'S5'><span texencoding="m_l=\sum{m_{i,p}} \ (i=1,2\ p=u,d)" style="vertical-align:-6px"><img src="" width="195" height="20.5" /></span><span>    (6.2)</span></div><div  class = 'S5'><span>通过插值拟合得到腿部质心位置和惯量关于 </span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">l</span><span> 的近似函数</span></div><div  class = 'S5'><span texencoding="(l_w,l_b)=f_{l,c}(l)" style="vertical-align:-6px"><img src="" width="89.5" height="19.5" /></span><span>    (6.3）</span></div><div  class = 'S5'><span texencoding="I_l=f_{l,I}(l)" style="vertical-align:-6px"><img src="" width="59.5" height="19.5" /></span><span>    (6.4)</span></div><h2  class = 'S4' id = 'H_53CCEAE5' ><span>附录1 运动学验证</span></h2>
<br>
<!-- 
##### SOURCE BEGIN #####
%% WBR腿部五连杆机构分析
%% 1 参数定义
% 
%% 2 逆运动学
% 逆运动学问题：$(\varphi_1,\varphi_2)=f(l,\theta)$
% 
% $l_1=\sqrt{l_{1,a}^2+l^2+2l_{1,a}lsin\theta}$    (2.1)
% 
% $l_2=\sqrt{l_{2,a}^2+l^2-2l_{2,a}lsin\theta}$    (2.2)
% 
% $\varphi_1=acos\frac{l_1^2+l_{1,a}^2-l^2}{2l_1l_{1,a}}+acos\frac{l_1^2+l_{1,u}^2-l_{1,d}^2}{2l_1l_{1,u}}$    
% (2.3)
% 
% $\varphi_2=acos\frac{l_2^2+l_{2,a}^2-l^2}{2l_2l_{2,a}}+acos\frac{l_2^2+l_{2,u}^2-l_{2,d}^2}{2l_2l_{2,u}}$    
% (2.4)
%% 3 正运动学
% 正运动学问题：$(l,\theta_l)=f^{-1}(\varphi_1,\varphi_2)$
% 
% $(l_{1,a}-l_{1,u}cos\varphi_1+lsin\theta)^2+(l_{1,u}sin\varphi_1-lcos\theta)^2=l_{1,d}^2$    
% (3.1)
% 
% $(l_{2,a}-l_{2,u}cos\varphi_2-lsin\theta)^2+(l_{2,u}sin\varphi_2-lcos\theta)^2=l_{2,d}^2$    
% (3.2)
% 
% 令 $x_e = lsin\theta$ , $y_e=lcos\theta$，记 $x_i=l_{i,a}-l_{i,u}cos\varphi_i$ 
% ,  $y_i = l_{i,u}sin\varphi_i$ , $i=1,2$
% 
% 方程组 (3.1)(3.2) 改写为
% 
% $(x_e+x_1)^2+(y_e-y_1)^2=l_{1,d}^2$    (3.3)
% 
% $(x_e-x_2)^2+(y_e-y_2)^2=l_{2,d}^2$    (3.4)
%% 
% 求解结果
% 
% $$x_e=\frac{y_{1}\sigma_1-y_{2}\sigma_1+{l_{1,d}}^2\,x_{1}+{l_{1,d}}^2\,x_{2}-{l_{2,d}}^2\,x_{1}-{l_{2,d}}^2\,x_{2}+x_{1}\,{x_{2}}^2-{x_{1}}^2\,x_{2}-x_{1}\,{y_{1}}^2-x_{1}\,{y_{2}}^2+x_{2}\,{y_{1}}^2+x_{2}\,{y_{2}}^2-{x_{1}}^3+{x_{2}}^3+2\,x_{1}\,y_{1}\,y_{2}-2\,x_{2}\,y_{1}\,y_{2}}{2\,\left({x_{1}}^2+2\,x_{1}\,x_{2}+{x_{2}}^2+{y_{1}}^2-2\,y_{1}\,y_{2}+{y_{2}}^2\right)}$$
% 
% $$y_e=\frac{x_{1}\sigma_1+x_{2}\sigma_1-{l_{1,d}}^2\,y_{1}+{l_{1,d}}^2\,y_{2}+{l_{2,d}}^2\,y_{1}-{l_{2,d}}^2\,y_{2}+{x_{1}}^2\,y_{1}+{x_{1}}^2\,y_{2}+{x_{2}}^2\,y_{1}+{x_{2}}^2\,y_{2}-y_{1}\,{y_{2}}^2-{y_{1}}^2\,y_{2}+{y_{1}}^3+{y_{2}}^3+2\,x_{1}\,x_{2}\,y_{1}+2\,x_{1}\,x_{2}\,y_{2}}{2\,\left({x_{1}}^2+2\,x_{1}\,x_{2}+{x_{2}}^2+{y_{1}}^2-2\,y_{1}\,y_{2}+{y_{2}}^2\right)}$$
% 
% 其中
% 
% $$\sigma_1=\sqrt{-{\left(-{l_{1,d} }^2 -2\,l_{1,d} \,l_{2,d} -{l_{2,d} }^2 
% +{x_1 }^2 +2\,x_1 \,x_2 +{x_2 }^2 +{y_1 }^2 -2\,y_1 \,y_2 +{y_2 }^2 \right)}\,{\left(-{l_{1,d} 
% }^2 +2\,l_{1,d} \,l_{2,d} -{l_{2,d} }^2 +{x_1 }^2 +2\,x_1 \,x_2 +{x_2 }^2 +{y_1 
% }^2 -2\,y_1 \,y_2 +{y_2 }^2 \right)}}$$
%% 4 雅可比矩阵
% 
% 
% 直接对式 (2.3)(2.4) 或方程组 (3.1)(3.2) 的解求偏导的结果非常复杂，而根据速度关系可以得到表达式相对简单的雅可比矩阵，即
% 
% $\dot{q}=J^{-1}\dot{p}$    (4.1)
% 
% 其中，$q=\left[\matrix{\varphi_1 & \varphi_2}\right]^T$，$p=\left[\matrix{l & 
% \theta}\right]^T$
% 
% 令 $x_e = lsin\theta$ , $y_e=lcos\theta$，记 $x_i=l_{i,a}-l_{i,u}cos\varphi_i$ 
% ,  $y_i = l_{i,u}sin\varphi_i$ , $i=1,2$
% 
% 速度约束方程
% 
% $\left(\dot\varphi_1l_{1,u}\left[\matrix{-sin\varphi_1 \cr cos\varphi_1}\right]\right)^T\left[\matrix{x_e+x_1 
% \cr y_e-y_1}\right]=\left(\dot l\left[\matrix{sin\theta \cr cos\theta}\right]+\dot\theta 
% l\left[\matrix{cos\theta \cr -sin\theta}\right]\right)^T\left[\matrix{x_e+x_1 
% \cr y_e-y_1}\right]$    (4.2)
% 
% $\left(\dot\varphi_2l_{2,u}\left[\matrix{sin\varphi_2 \cr cos\varphi_2}\right]\right)^T\left[\matrix{x_e-x_2 
% \cr y_e-y_2}\right]=\left(\dot l\left[\matrix{sin\theta \cr cos\theta}\right]+\dot\theta 
% l\left[\matrix{cos\theta \cr -sin\theta}\right]\right)^T\left[\matrix{x_e-x_2 
% \cr y_e-y_2}\right]$    (4.3)
% 
% 由式 (4.2)(4.3) 可得
% 
% $\frac{\partial\varphi_1}{\partial l}=\frac{1}{l_{1,u}}\frac{(x_e+x_1)sin\theta+(y_e-y_1)cos\theta}{-(x_e+x_1)sin\varphi_1+(y_e-y_1)cos\varphi_1}$    
% (4.4)
% 
% $\frac{\partial\varphi_1}{\partial\theta}=\frac{l}{l_{1,u}}\frac{(x_e+x_1)cos\theta-(y_e-y_1)sin\theta}{-(x_e+x_1)sin\varphi_1+(y_e-y_1)cos\varphi_1}$    
% (4.5)
% 
% $\frac{\partial\varphi_2}{\partial l}=\frac{1}{l_{2,u}}\frac{(x_e-x_2)sin\theta+(y_e-y_2)cos\theta}{(x_e-x_2)sin\varphi_2+(y_e-y_2)cos\varphi_2}$    
% (4.6)
% 
% $\frac{\partial\varphi_2}{\partial\theta}=\frac{l}{l_{2,u}}\frac{(x_e-x_2)cos\theta-(y_e-y_2)sin\theta}{(x_e-x_2)sin\varphi_2+(y_e-y_2)cos\varphi_2}$    
% (4.7)
% 
% $J=\left[\matrix{\frac{\partial\varphi_1}{\partial l} & \frac{\partial\varphi_1}{\partial\theta} 
% \cr \frac{\partial\varphi_2}{\partial l} & \frac{\partial\varphi_2}{\partial\theta}}\right]^{-1}$    
% (4.8)
%% 5 动态静力学
% 
% 
% 记 $\gamma=\left[\matrix{F_{bl} & T_{bl}}\right]^T$，$\tau=\left[\matrix{\tau_1 
% & \tau_2}\right]^T$
% 
% 根据虚功原理可得
% 
% $\tau=J^T\gamma$    (5.1)
%% 6 动力学参数
% $\theta_l=\theta+\theta_b$    (6.1)
% 
% $m_l=\sum{m_{i,p}} \ (i=1,2\ p=u,d)$    (6.2)
% 
% 通过插值拟合得到腿部质心位置和惯量关于 $l$ 的近似函数
% 
% $(l_w,l_b)=f_{l,c}(l)$    (6.3）
% 
% $I_l=f_{l,I}(l)$    (6.4)
%% 附录1 运动学验证
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